According to the Angular docs
min (optional) string Sets the min validation error key if the value
entered is less than min. This must be a valid ISO date string
(yyyy-MM-dd).
you can bind that to a scope variable or directly put the expression in the attribute, it just needs to be in the correct format (javascript date object has a toISOString() method). It should also be noted this does not prevent the user from picking a date below the minimum and instead sets the validation key to indicate the field is not valid.
edit:
Script
var app = angular.module('plunker', []);
app.controller('MainCtrl', function($scope) {
$scope.name = 'World';
var today=new Date();
$scope.today = today.toISOString();
});
Html
<!DOCTYPE html>
<html ng-app="plunker">
<head>
<meta charset="utf-8" />
<title>AngularJS Plunker</title>
<script>document.write('<base href="' + document.location + '" />');</script>
<link rel="stylesheet" href="style.css" />
<script data-require="angular.js@1.3.x" src="https://code.angularjs.org/1.3.14/angular.js" data-semver="1.3.14"></script>
<script src="app.js"></script>
</head>
<body ng-controller="MainCtrl">
<p>Hello {{name}}!</p>
<form name="myForm">
<input type="date" ng-model="StartDate" name="StartDate" required min="{{today}}" />
<span ng-show="myForm.StartDate.$dirty && myForm.StartDate.$invalid">
<span ng-show="myForm.StartDate.$error.required">Start Date is required</span>
<span ng-show="myForm.StartDate.$error.min">Start Date should not be less than current date</span>
</span>
</form>
</body>
</html>
your error message indicates you want the date to be no greater than the current date, if that is the case simply change
<input type="date" ng-model="StartDate" name="StartDate" required min="{{today}}" />
to
<input type="date" ng-model="StartDate" name="StartDate" required max="{{today}}" />
Working Example